DBAR problem

The DBAR problem, or the ${\displaystyle {\bar {\partial }}}$-problem, is the problem of solving the differential equation

${\displaystyle {\bar {\partial }}f(z,{\bar {z}})=g(z)}$

for the function ${\displaystyle f(z,{\bar {z}})}$, where ${\displaystyle g(z)}$ is assumed to be known and ${\displaystyle z=x+iy}$ is a complex number in a domain ${\displaystyle R\subseteq \mathbb {C} }$. The operator ${\displaystyle {\bar {\partial }}}$ is called the DBAR operator

${\displaystyle {\bar {\partial }}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right)}$

The DBAR operator is nothing other than the complex conjugate of the operator

${\displaystyle \partial ={\frac {\partial }{\partial z}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}\right)}$

denoting the usual differentiation in the complex ${\displaystyle z}$-plane.

The DBAR problem is of key importance in the theory of integrable systems[1] and generalizes the Riemann–Hilbert problem.

References

1. Konopelchenko, B. G. "On dbar-problem and integrable equations". arXiv:nlin/0002049.